Under ZFC, the real numbers can be well-ordered. So, there is some ordinal number whose cardinality is that of the continuum. Is there a standard notation for this number?
For example, the first infinite ordinal is usually denoted $\omega$, and the first uncountable ordinal is usually denoted $\omega_1$. But unless we appeal the continuum hypothesis (or something just as presumptuous), $\omega_1$ may not have cardinality of the continuum.
Here is a related question, which is just my curiosity at work: Do we know whether there is such thing as a least and/or greatest ordinal with cardinality of the continuum?
Thanks!
P.S. One last recreational-math question: Are there any other well-known measures of a number's size besides ordinals and cardinals?
Not really. If one assumes the axiom of choice, as one usually does, then one has that $2^{\aleph_0}$ is in fact an ordinal, an initial ordinal. So writing $\alpha<2^{\aleph_0}$ has a very clear meaning. To that end, using $\frak c$ is also quite common as denoting the cardinality of the continuum. So it is not unusual to see $\alpha<\frak c$ in set theoretical papers.
So yes, there is such a least ordinal, this is the initial ordinal (by the definition of an initial ordinal). There is no maximal ordinal, just like there is no "largest countable ordinal". If $X$ is an infinite set, there is no largest ordinal equipotent to $X$. In fact, there are exactly $|X|^+$ ordinals which are equipotent to $X$. Just like there are $\aleph_1$ countably infinite ordinals, and $\aleph_2$ ordinals of size $\aleph_1$, and so on.